Manuscript Hawk03-A@1 · Hawk03-A.tex · lemma · 9a109c12

Lemma: Exact encoded instrument

Encoded algebras and the no-independent-copy obstruction · lines 717–741 · proof included or linked

Hawk03-A@1#lem:instrument

Suppose $J$ is unitary from $H_{L}$ onto the physical space and ${L_{x, a}}$ are Kraus operators of a complete logical instrument. Then

L_{x, a} = J L_{x, a} J^{†}

define a complete physical instrument and satisfy

I_{x} (J ρ J^{†}) = J I_{x} (ρ) J^{†} .

The identity holds with every untouched reference and under any finite adaptive composition of such instruments.

Proof. Unitary conjugation preserves $\sum_{x, a} L_{x, a}^{†} L_{x, a} = I$ and complete positivity. The displayed channel identity follows by multiplication. Tensoring all operators with $I_{Q}$ proves the reference version. Induction on unnormalized conditional states proves the adaptive version, including all failure outcomes.

Displayed formulas in this passage

unlabeled · line 722
$L_{x, a} = J L_{x, a} J^{†}$
unlabeled · line 726
$I_{x} (J ρ J^{†}) = J I_{x} (ρ) J^{†} .$

Lemma: Exact encoded instrument

Displayed formulas in this passage

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